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in the plane of the rotor. For the articulated rotor, a drag hinge is provided to relieve
0.30
0.25
0.20
0.15
0.10
0.05
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
30 25 20 15 % chordwise c.g.
Stable
Unstable
Coning angle, β0 radians
A A 1/K / ′ ⋅ ′θ
B/A = 1000
IB = 1, θ0 = 0
Fig. 9.12 Stability boundaries of blade weaving motion
336 Bramwell’s Helicopter Dynamics
the blade of these moments and allow it to move in the plane of rotation (lagging).
For semi-rigid and totally bearingless rotors, and for the majority of tail rotors, the
lagwise flexibility provided at the root end of the blade permits the corresponding
lagwise movement, and is designed to accommodate the resulting moments.
The lagging motion means that the instantaneous angular velocity of the blade in
the plane of rotation is slightly different from the (assumed) constant angular velocity
of the shaft, and this in turn means that the centrifugal flapping moment depends on
the lagging motion. The variations of relative airspeed due to lagging also affect the
aerodynamic flapping moment. Thus, the flapping and lagging motions are clearly
coupled, but to investigate the stability of these motions we must derive the appropriate
equations of motion. Referring to Fig. 9.14, if the lagging angular velocity is ˙ ξ , then
the instantaneous angular velocity is Ω + ˙ ξ and, by the arguments of Chapter 1, the
centrifugal flapping moment is –B(Ω + ˙ ξ )2β, tending to restore the blade to the plane
of rotation. Neglecting the term in ˙ξ2, the first-order flapping moment is
–B(Ω2 + 2Ω ˙ ξ ) β and the equations of flapping and lagging motion are
Bβ˙˙ + Bλ 2β + 2B βξ˙ = M
1 A
2Ω Ω (9.19)
Total damping
Total control circuit stiffness
Unstable
Fig. 9.13 Stability boundary for coupled pitch–flap motion
Aeroelastic and aeromechanical behaviour 337
Ω
β
ξ
Fig. 9.14 Simple flap–lag blade model
Cξ˙˙ + Cκ 2ξ – 2C ββ˙ = N
1 A
2Ω Ω (9.20)
where we have written λ1Ω and κ1Ω for the undamped natural flapping and lagging
frequencies which can apply to both hinged and hingeless blades. The aerodynamic
moments MA and NA will contain aerodynamic coupling terms. To calculate them we
consider the force on a blade element under conditions of flapping and lagging. For
simplicity we consider only hovering flight. Referring to Fig. 9.15, the elementary
flapping and lagging forces are, respectively,
dZ = dL cos φ + dD sin φ ≈ dL
dY = dL sin φ dD cos φ ≈ dLφ – dD
Now
d = ( + )d 12
L ρW2 caθ φ r
≈ ( + ) d 12
T
ρU2 caθ φ r
and
d d 12
T
D ≈ ρU2 cCD r
Also,
tan φ ≈ φ = UP/UT
dL dZ
UT
UP
W
dD
φ dY
θ
φ
Fig. 9.15 Forces on blade element
338 Bramwell’s Helicopter Dynamics
therefore,
d = ( + ) 12
T
2
Z ρac dr θU UPUT (9.21)
and
d = ( + ) – d 12
P T P
2 12
T
Y ρac dr θU U U ρU2CDc r (9.22)
The velocity components UP and UT are
U T = (Ω + ξ˙)r; UP = – rβ˙ – vi
Expanding eqns 9.21 and 9.22 and neglecting squares and products of ˙β and ˙ξ
gives
d = [ ( + 2 ) – – – ]d 12
2 2 2 2
Z ρacθΩr Ωξ˙r Ωβ˙r Ωrvi Ωξ˙vi r
d = – ( + 2 )d 12
Y ρcCDΩ2r2 Ωξ˙r2 r
– ( + + – – )d 12
2
i i i
2
ρacθΩβ˙r θΩrv θξ˙rv v 2β˙rvi r
Integrating rdZ and rdY, assuming CD, θ, and vi to be constant, we obtain for the
aerodynamic flapping and lagging moments
MA ac R
2 4
= 1 i i
8
– 4
3
– + 2 – 4
3
ρ Ω θ λ Ωβ θ λ Ωξ
˙ ˙
(9.23)
N ac R
C
a
D
A
2 4
i i
2
= – 1 i
8
+ 4
3
– 2 + – 8
3
ρΩ θλ λ θ λ Ωβ
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